https://file.aiquickdraw.com/mathbot/file/1726883690960s8l2p742.pdf

The document contains a set of remedial trigonometric limit problems. Here are the questions extracted from the document:

### 1. Evaluate:
\[
\lim_{x \to 0} \frac{\sin(12x)}{2x^2 + 3} - 2
\]
Options:  
a. –4, b. –3, c. –2, d. 2, e. 6

### 2. Evaluate:
\[
\lim_{x \to 0} \frac{2 \cos(x)}{1 - 2 \sin(2x)}
\]
Options:  
a. \( \frac{1}{8} \), b. \( \frac{1}{6} \), c. \( \frac{1}{4} \), d. \( \frac{1}{2} \), e. 1

### 3. Evaluate:
\[
\lim_{x \to 0} \frac{1 - \cos(4x)}{1 - \cos(2x)}
\]
Options:  
a. \( -\frac{1}{2} \), b. \( -\frac{1}{4} \), c. 0, d. \( \frac{1}{16} \), e. \( \frac{1}{4} \)

### 4. Evaluate:
\[
\lim_{x \to 0} \frac{\sin(5x) + \sin(6x)}{x}
\]
Options:  
a. 2, b. 1, c. \( \frac{1}{2} \), d. \( \frac{1}{3} \), e. –1

### 5. Evaluate:
\[
\lim_{x \to 0} \frac{1 - 2 \cos(x)}{\tan^3(x)}
\]
Options:  
a. \( \frac{8}{9} \), b. \( \frac{2}{9} \), c. \( \frac{1}{9} \), d. 0, e. \( -\frac{6}{9} \)

Would you like solutions to any of these questions? Feel free to ask for details or any clarifications.

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Here are five related questions to expand your understanding:

1. How do you apply L'Hopital's rule to trigonometric limits?
2. What is the limit of \( \frac{\sin x}{x} \) as \( x \to 0 \), and why is it useful in trigonometric limit problems?
3. How can Taylor series be used to approximate limits involving trigonometric functions?
4. What are common indeterminate forms encountered in limit problems?
5. How does the small-angle approximation for sine and cosine functions help in solving limits?

**Tip:** Always check for indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) before applying L'Hopital's Rule.

The document contains a set of remedial trigonometric limit problems. Here are the questions: 1. lim(x→0) [(sin(12x))/(2x^2 + 3)] - 2, 2. lim(x→0) [(2 cos(x))/(1 - 2 sin(2x))], 3. lim(x→0) [(1 - cos(4x))/(1 - cos(2x))], 4. lim(x→0) [(sin(5x) + sin(6x))/x], 5. lim(x→0) [(1 - 2 cos(x))/(tan^3(x))].

Explore five challenging trigonometric limit problems involving L'Hopital's Rule and small-angle approximations. Learn how to solve limits like lim(x→0) [(sin(12x))/(2x^2 + 3)] - 2 and others using key trigonometric formulas and concepts. Suitable for Grades 11-12 students.

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